Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms
Affiliation(s)
The National Academy of Science, Seoul, Korea. Hachinohe National College of Technology, Aomori, Japan.
The National Academy of Science, Seoul, Korea. Hachinohe National College of Technology, Aomori, Japan.
ABSTRACT
Let M be a real hypersurface of a complex space form with almost contact metric structure (φ,ξ,η,g). In this paper, we prove that if the structure Jacobi operator Rξ=(·,ξ) ξ is φ▽ξξ-parallel and Rξ commute with the shape operator, then M is a Hopf hypersurface. Further, if Rξ is φ▽ξξ-parallel and Rξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrRξ is constant.
KEYWORDS
Complex Space Form; Hopf Hypersurface; Structure Jacobi Operator; Shape Operator; Ricci Tensor
Complex Space Form; Hopf Hypersurface; Structure Jacobi Operator; Shape Operator; Ricci Tensor
Cite this paper
U. Ki and H. Kurihara, "Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms," Advances in Pure Mathematics, Vol. 3 No. 2, 2013, pp. 264-276. doi: 10.4236/apm.2013.32038.
U. Ki and H. Kurihara, "Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms," Advances in Pure Mathematics, Vol. 3 No. 2, 2013, pp. 264-276. doi: 10.4236/apm.2013.32038.
References
[1] T. E. Cecil and P. J. Ryan, “Focal Sets and Real Hypersurfaces in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 269, No. 2, 1982, pp. 481-499. [2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 19, 1973, pp. 495-506. [3] R. Takagi, “Real Hypersurfaces in a Complex Projective Space with Constant Principal Curvatures I, II,” Journal of the Mathematical Society of Japan, Vol. 15, No. 43-53, 1975, pp. 507-516. [4] M. Kimura, “Real Hypersurfaces and Complex Submanifolds in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 296, No. 1, 1986, pp. 137-149. [5] J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperblic Spaces,” Journal für die Reine und Angewandte Mathematik, Vol. 395, 1989, pp. 132-141. [6] J. T. Cho and U-H. Ki, “Jacobi Operators on Real Hypersurfaces of a Complex Projective Space,” Tsukuba Journal of Mathematics, Vol. 22, 1988, pp. 145-156. [7] J. T. Cho and U-H. Ki, “Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Jacobi Operator,” Canadian Mathematical Bulletin, Vol. 51, No. 3, 2008, pp. 359-371. [8] U-H. Ki, H. Kurihara, S. Nagai and R. Takagi, “Characterizations of Real Hypersurfaces of Type A in a Complex Space Form in Terms of the Structure Jacobi Opera- tor,” Mathematics Journal of Toyama University, Vol. 32, 2009, pp. 5-23. [9] J. D. Pérez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Nonflat Complex Space Forms with Commuting Structure Jacobi Operator,” Houston Journal of Mathematics, Vol. 33, 2007, pp. 1005-1009. [10] M. Ortega, J. D. Pérez and F. G. Santos, “Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms,” Rocky Mountain Journal of Mathematics, Vol. 36, No. 5, 2006, pp. 1603-1613. [11] J. D. Pérez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Spaces Whose Structure Jacobi Operator is D-Parallel,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 13, No. 3, 2006, pp. 459-469. [12] U.-H. Ki, H. Kurihara and R. Takagi, “Jacobi Operators along the Structure Flow on Real Hypersurfaces in a Nonflat Complex Space Form,” Mathematics Journal of Toyama University, Vol. 33, 2009, pp. 39-56. [13] U.-H. Ki and H. Kurihara, “Real Hypersurfaces and ξ-Parallel Structure Jacobi Operators in Complex Space Forms,” Journal of Korean Academy Sciences, Sciences Series, Vol. 48, 2009, pp. 53-78.
[1] T. E. Cecil and P. J. Ryan, “Focal Sets and Real Hypersurfaces in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 269, No. 2, 1982, pp. 481-499. [2] R. Takagi, “On Homogeneous Real Hypersurfaces in a Complex Projective Space,” Osaka Journal of Mathematics, Vol. 19, 1973, pp. 495-506. [3] R. Takagi, “Real Hypersurfaces in a Complex Projective Space with Constant Principal Curvatures I, II,” Journal of the Mathematical Society of Japan, Vol. 15, No. 43-53, 1975, pp. 507-516. [4] M. Kimura, “Real Hypersurfaces and Complex Submanifolds in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 296, No. 1, 1986, pp. 137-149. [5] J. Berndt, “Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperblic Spaces,” Journal für die Reine und Angewandte Mathematik, Vol. 395, 1989, pp. 132-141. [6] J. T. Cho and U-H. Ki, “Jacobi Operators on Real Hypersurfaces of a Complex Projective Space,” Tsukuba Journal of Mathematics, Vol. 22, 1988, pp. 145-156. [7] J. T. Cho and U-H. Ki, “Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Jacobi Operator,” Canadian Mathematical Bulletin, Vol. 51, No. 3, 2008, pp. 359-371. [8] U-H. Ki, H. Kurihara, S. Nagai and R. Takagi, “Characterizations of Real Hypersurfaces of Type A in a Complex Space Form in Terms of the Structure Jacobi Opera- tor,” Mathematics Journal of Toyama University, Vol. 32, 2009, pp. 5-23. [9] J. D. Pérez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Nonflat Complex Space Forms with Commuting Structure Jacobi Operator,” Houston Journal of Mathematics, Vol. 33, 2007, pp. 1005-1009. [10] M. Ortega, J. D. Pérez and F. G. Santos, “Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms,” Rocky Mountain Journal of Mathematics, Vol. 36, No. 5, 2006, pp. 1603-1613. [11] J. D. Pérez, F. G. Santos and Y. J. Suh, “Real Hypersurfaces in Complex Projective Spaces Whose Structure Jacobi Operator is D-Parallel,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 13, No. 3, 2006, pp. 459-469. [12] U.-H. Ki, H. Kurihara and R. Takagi, “Jacobi Operators along the Structure Flow on Real Hypersurfaces in a Nonflat Complex Space Form,” Mathematics Journal of Toyama University, Vol. 33, 2009, pp. 39-56. [13] U.-H. Ki and H. Kurihara, “Real Hypersurfaces and ξ-Parallel Structure Jacobi Operators in Complex Space Forms,” Journal of Korean Academy Sciences, Sciences Series, Vol. 48, 2009, pp. 53-78.